Digital Signal Processing Reference
In-Depth Information
300
100
L
internal
90
250
80
70
200
R
ac
60
150
50
40
100
30
20
50
10
0
0
1.00E
+
07
1.00E
+
08
1.00E
+
09
1.00E
+
10
1.00E
+
11
Frequency, Hz
Figure 5-11
Example of how the skin effect changes resistance and internal inductance
with frequency for a copper microstrip as shown in Figure 5-6.
the skin depth in copper at 2 GHz using (5-10) and compare it to the conductor
thickness
t
:
2
2
πf µ
0
σ
≈
δ
=
1
.
41
m
µ
Since
t
=
m,
δ<t
, so the ac resistance must be used.
Step 2:
The impedance and effective dielectric permittivity must be calculated
using equation (3-36b), which assumes a perfect conductor. If it is unknown
whether a particular impedance formula includes the effect of a realistic conduc-
tor, the lack of a metal conductivity or a magnetic permeability variable indicates
the assumption of infinite conductivity. From (3-36b),
Z
0
0
.
5 mil
=
12
.
7
µ
≈
55
and
ε
eff
≈
2
.
95.
Step 3:
Calculate the phase velocity with (2-52):
10
8
m/s
√
(
1
)(
2
.
95
)
=
c
√
µ
r
ε
eff
3
×
10
8
m/s
ν
p
=
=
1
.
75
×
Step 4:
The external inductance is solved with (3-31) and (3-33). Since (3-36b)
assumes a perfect conductor, the inductance is the external value.
→
√
LC
=
1
√
LC
=
10
8
10
−
9
ν
p
=
1
.
75
×
5
.
73
×
s/m
10
−
12
≈
146
×
s/in.
L
C
=
Z
0
=
55
L
C
·
√
LC
=
10
−
12
)
=
L
external
=
L
=
55
(
146
×
8
.
03 nH/in.
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